# Each integer appears once in difference sequence

Does there exist an increasing sequence $a_1,a_2,\dots$ of positive integers such that both of the following are fulfilled:

• Each positive integer appears exactly once among $a_2-a_1,a_3-a_2,\dots$

• For some $n$, each positive integer at least $n$ appears exactly once among $a_3-a_1,a_4-a_2,a_5-a_3,\dots$?

If we only require the first condition, we can take the sequence $1,2,4,7,11,\dots$, while if we only require the second condition, we can construct the sequence by induction, making sure that each difference appears once.

• Are you asking if it can be constructed by induction? Or are you saying it can be constructed by induction? Sep 12 '16 at 9:33
• I'm saying that if we only require the second condition, the sequence can be constructed by induction.
– pi66
Sep 12 '16 at 9:34
• So your question is, can both the conditions be satisfied together? Because the question is not clear. Sep 12 '16 at 9:35
• "such that both of the following are fulfilled"
– pi66
Sep 12 '16 at 9:36
• Ok great. I'll have a look. Sep 12 '16 at 9:36

To fulfil the first condition, the differences $a_i-a_{i-1}$ must be distinct positive integers. Thus, to avoid repeating a difference, $$a_n\ge a_0+\frac{n(n+1)}2.$$ The second condition requires that all but finitely many positive integers appear among the $a_i-a_{i-2}$. But the above growth is too fast to enable that. Thus the second condition cannot be fulfilled.